my answers are for Neptune specifically, rather than for any liquid cored planet, though many principles used are generalise-able.
on a "small scale": you would need to use robots to physically go down and collect the liquid in the core from the gaseous atmosphere. there are two approaches to this. firstly you can make a robot-plane able to go all the way from the 'surface' to the deep high pressure centre, alternatively a sort of relay system could be made so many robots pass the liquid between each other each staying within their own band of pressure and altitude with an internal pressure to match. the relay goes all the way from the bottom to the top though no one robot does. both are viable and doable, though I strongly recommend the second.
on large scales: use a "centrifugal straw". the internal pressures in Neptune are a massive problem, however the temperatures are apparently not (being significantly under 5100k which while extremely hot is kind of workable with something like graphite since the high pressures raise the boiling point). back to pressure their is fortunately a way for a pipe to overcome arbitrarily high pressures. spinning the pipe (or rather an internal "support" in the walls of the pipe) will counter the inward force of the atmospheric pressure. since you can spin something arbitrarily quickly there is no theoretical limit to the pressure such a system can be built to withstand. the pipe could not be a suction pipe (https://www.youtube.com/watch?v=HUmZrtiXDik) and would need to support its own weight using buoyancy. power would be provided from large solar arrays in orbit or at the surface.
sadly the only information i could get on neptunes core temperature is this very uncited image that looks like its from 2003 but that being said I will still be using it
on a very large scale: like eggs in a microwave! the core is big meaning it has a large volume to surface area ratio so if parts of it are heated they will keep that heat for a vary long time. so using an oscillating magnetic field, radiation or some kind of high pressure tolerance nuclear bombs dropped deep into the core you could heat the core up substantially. hot matter rises (admittedly you need to make it very hot if its going to rise through hydrogen and helium but its still doable especially given how cold neptunes atmosphere is) so large thermals of the core substances will start blooming in the atmosphere for 'easy mining'. odd but doable for any K1.(a bit) civilisation.
on an absurdly large scale: just mine the whole planet layer by layer with a vast legion of robots until you reach the juicy core. I mean why not? you were probably going to be mining the outer layers anyway, right?
